AP CALCULUS AB SECTION I, Part A Time 55 Minutes Number of questions 28 Time Began: Time Ended: A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM Directions: Solve each of the following problems, using the available space for scratchwork. Do not spend too much time on any one problem. In this assignment: 1. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which is a real number. 2. The inverse of a trigonometric function f may be indicated using the inverse function notation or with the prefix arc (e.g. ). 1. (A) 10 40 80 5x 2 3x + 1 2. lim x 4x 2 + 2x + 5 is (A) 0 4 5 3 11 5 4 3. (A) 1
4. If the function f is continuous for all real number and if f x ( ) = x2 7x + 12 x 4 when x 4, then f ( 4) = (A) 1 0 undefined 5. (A) 6. Which of the following integrals correctly corresponds to the area of the shaded region in the figure above? (A)
7. (A) 0 8. An equation of the line normal to the graph of y = 3x 2 + 2x at is (A) 9. 1 4 1 1 + x dx = 2 (A) 0 1 2 10. If f ( x) = cos 2 x, then f "( π ) = (A) -2 0 1 2
11. (A) 1 ( 10 5x2 4) 3 2 + C ( ) 3 2 + C 20 3 5x2 4 1 ( 15 5x2 4) 3 2 + C 1 ( 5 5x2 4) 3 2 + C ( ) 3 2 + C 3 20 5x2 4 12. The slope of the line tangent to the graph of at is (A) 12 13. If f x, for all real number x, which of the following must be true 7x 5 if x 2 ( ) = x2 + 5 if x < 2 I. f x II. f x III. f x ( ) is continuous everywhere. ( ) is differentiable everywhere. ( ) has a local minimum at x = 2. (A) I only I and II only II and III only I and III only I, II, and III 14. The graph of a piecewise linear function f, for, is shown above. What is the value of? (A) 1 4 8 10 13
15. If, then (A) 16. What is the instantaneous rate of change at of the function f, if f ( t) = t 3 + t 4t + 1? (A) 12 9 4 9 20 9 4 9 12 9 17. (A) 4 4e 0 18. lim h 0 tan π 6 + h tan π 6 h = (A) 0
19. The average value of the function f ( x) = ( x 1) 2 on the interval from to is (A) 16 3 16 3 64 3 ` 66 3 256 3 20. A solid is generated when the region in the first quadrant enclosed by the graph of, the line, the x- axis, and the y-axis is revolved about the x-axis. Its volume is found by evaluating which of the following integrals? (A) 21. If and when, then when, (A) 18 58
22. If then (A) 0 23. A particle moves along the x-axis so that its position at time t, in seconds, is given by. For what value(s) of t is the velocity of the particle zero? (A) 1 6 1 or 6 3.5 1 or 3.5 or 6 24. If, then (A) 0 25. Find the area of the region bounded by the parabolas and. (A) 9 27 6
26. The function f is given by On which intervals is f decreasing? (A) ( 3,0) (, 3) 27. Find the value of c that satisfies the Mean Value Theorem on the interval for the function (A) 0 1
28. The graph of is shown in the figure above. Which of the following could be the graph of. STOP END OF PART A SECTION 1
AP CALCULUS AB SECTION I, Part B Time 50 Minutes Number of Questions 17 Time Began: Time Ended: A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM Directions: Solve each of the following problems, using the available space for your work. Place your answer on the line in the bottom right hand corner of the space. Do not spend too much time on any one problem In this assignment: 1. The exact numerical value of a correct answer may not always be found. Your answer should then be given as the best approximation, correct to three decimal places. 2. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which is a real number. 3. The inverse of a trigonometric function f may be indicated using the inverse function notation or with the prefix arc (e.g. ). 0 4 76. sin x dx + cos x dx = 0 π π 4 (A) 2 0 1 2 77. d dx tan2 ( 4x) = (A)
78. (A) 0 20 79. If, and, then could be (A) 59 11 80. The rate at which water is sprayed on a field of vegetables is given by R( t) = 2 1 + 5t 3, where t is in minutes at per minute? is in gallons per minute. During the time interval, what is the average rate of water flow, in gallons (A) 8.458 13.395 14.691 18.916 35.833 81. If the function f x is continuous and differentiable at all real numbers, then a = bx 2 + 4; if x > 1 ( ) = ax3 6x; if x 1 (A) 0 1 14 24 26
82. (A) tan 7 x + C 7 tan 7 x 7 + sec3 x 3 tan 7 x sec 3 x + C 21 + C 7 tan 7 x + C 2 7 tan7 x sec x + C 83. A 20 foot ladder slides down a wall at 5 ft/sec. At what rate is the bottom of the ladder sliding out when the top is 10 feet from the floor? (A) 0.346 2.887 0.224 5.774 4.472 84. Find the distance traveled (to three decimals ) from to seconds, for a particle whose velocity is given by. (A) 6.000 1.609 16.047 0.800 148.413
85. : x f g f g a -4 c 8 b c 15 10 6 5 If, find. (A) 80 86. The second derivative of a function f is given by. How many points of inflection does f have on the interval? (A) Zero Two Four Six Eight 87. Find the total area of the region between the curve and the x-axis from to radians. (A) 0 0.068 0.249 1.751 2.592 88. The tangent line to the curve at the point has an x-intercept at what point? (A)
89. The graph of, the derivative of f, is shown above. On which interval(s) is the graph of f decreasing? (A) 90. A particle moves along the x-axis so that at any time its velocity is given by. What is the acceleration of the particle at time? (A) 1.500 20.453 29.453 74.860 133.417
91. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of, each using the same number of subintervals. The graph of the function f is shown in the figure above. Which of the sums give an underestimate of the value? I. Left sum II. Right sum III. Trapezoidal sum (A) I only II only III only I and III only II and III only 92. Let f be the function with the first derivative defined by for. At what value of x does f attain its absolute maximum value on the closed interval? (A) 0 1.162 1.468 1.845 2 STOP END OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO TO SECTION II UNTIL YOU ARE TOLD TO DO SO.